MODELING OF A DISTILLATION COLUMN

USING BOND GRAPHS

by

Braden Alan Brooks

A Thesis Submitted to the Faculty of the

Department of Electrical & Computer Engineering

In Partial Fulfillment of the Requirements For the Degree of

MASTER OF SCIENCE WITH A MAJOR IN ELECTRICAL ENGINEERING

in the Graduate College

THE UNIVERSITY OF ARIZONA

Copyright © Braden Alan Brooks 1993

1 9 9 3

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STATEMENT BY AUTHOR

This thesis has been submitted in partial fulfillment of requirements for an advanced degree at the University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.

Brief quotations from this thesis are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the copyright holder. SIGNED:

APPROVAL BY THESIS DIRECTOR

This thesis has been approved on the date shown below: François E. Cellier Date Associate Professor of Electrical and Computer Engineering

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ACKNOWLEDGMENT

To let understanding stop at what cannot be understood is a high attainment. Those who cannot do this will be destroyed on the lathe of heaven.

Tchuang-tse

The author would like to thank the universe for not being as perversely

ordered as once thought.

Also, considerable thanks goes to the patience and faith of my advisor,

Dr. François Cellier, and to my parents, Billy and Jean Brooks.

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TABLE OF CONTENTS

LIST OF FIGURES.......................................................................................................6

LIST OF TABLES.........................................................................................................9

ABSTRACT ..................................................................................................................10

1.0 INTRODUCTION ................................................................................................11 1.1 The basics of a distillation column ........................................................11 1.2 The basics of bond graphs ......................................................................16 1.3 The assumptions of a distillation column ............................................19

2.0 THE ELEMENTS OF A DISTILLATION COLUMN ......................................22 2.1 Mass balance.............................................................................................22 2.2 Energy balance .........................................................................................25 2.3 Hydraulic (pressure and volume flow) ................................................28 2.4 Vapor-liquid equilibrium (chemical potential and molar flow) .......42 2.5 Heat (temperature and entropy)............................................................50 2.6 Equation of state.......................................................................................52 2.7 Condenser and receiver ..........................................................................56 2.8 Reboiler......................................................................................................62 2.9 Control system..........................................................................................68

3.0 MODELING WITH BOND GRAPHS ...............................................................70 3.1 Bond graph representation of the model..............................................70

3.1.1 Hydraulic ....................................................................................72 3.1.2 Thermal........................................................................................78 3.1.3 Chemical......................................................................................82 3.1.4 Combining hydraulic, thermal, and chemical .......................84

3.2 Hierarchical bond graph representation ..............................................88

4.0 THE DISTILLATION COLUMN AS A SYSTEM ............................................94 4.1 The early models ......................................................................................94 4.2 The Gallun model ....................................................................................96 4.3 Modifying the Gallun model..................................................................102 4.4 Simulation results ....................................................................................106

5.0 CONCLUSION.....................................................................................................115 5.1 Future work in distillation.......................................................................115

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5.2 Future work in bond graphs ...................................................................117

TABLE OF CONTENTS — Continued

APPENDIX A: NOTATION .....................................................................................121

APPENDIX B: GALLUN’S DISTILLATION COLUMN .......................................128 B.1 The equations of the model.....................................................................128 B.2 DYMOLA code of Gallun’s model.........................................................136 B.3 DYMOLA assessment of Gallun’s model .............................................159

APPENDIX C: ACSL CODE FOR SIMPLE DISTILLATION COLUMN MODEL..............................................................................................167

APPENDIX D: ACSL CODE FOR COMPLEX DISTILLATION COLUMN MODEL..............................................................................................188

REFERENCES..............................................................................................................213

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LIST OF FIGURES

1.1. The basic stage of a distillation column...........................................................12

1.2. The basic distillation column.............................................................................13

2.1. Bond graph of liquid mass balance equation (2-8).........................................25

2.2. Internal energy elements....................................................................................28

2.3. Bond graph of fluid flow, equation (2-28) .......................................................33

2.4. The hydraulic variables in a distillation column stage..................................34

2.5. Bond graph of downcomer hydraulics, equation (2-31) ...............................36

2.6. Bond graphs of Francis weir formula, .............................................................40

2.7. Transformer representing chemical to hydraulic power conversion..........45

2.8. The bond graph of the condenser, equations (2-70).......................................58

2.9. Cooling water flow in condenser......................................................................59

2.10. Bond graph of pump ........................................................................................60

2.11. Bond graph of pump and valves below receiver .........................................60

2.12. Bond graphs of pump and valves without an algebraic loop ....................61

2.13. HOV element .....................................................................................................63

2.14. Thermal and hydraulic bond graph of reboiler............................................65

2.15. Hydraulic bond graph of the base of the column ........................................67

2.16. Bond graph of PI control of valves .................................................................69

3.1. Convection bond graph of distillation stage...................................................71

3.2. Bond graph of liquid hydraulics.......................................................................72

3.3. Bond graph of vapor hydraulics.......................................................................73

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LIST OF FIGURES — Continued

3.4. Expanded completion to hydraulic bond graph ............................................73

3.5. Fugacity driven phase flow of traditional model...........................................75

3.6. Hydraulic bond graph of distillation stage .....................................................77

3.7. Bond graph of thermal power flow..................................................................79

3.8. Chemical bond graph of distillation stage ......................................................82

3.9. Bond graph of distillation stage........................................................................87

3.10. Bond graph of fluid flow represented by the Fluid element ......................88

3.11. Tray bond graph represented by the Foam element....................................89

3.12. Bond graph associated with the feed represented by Feed element .........89

3.13. Distillation stage bond graph represented by the DCStage element ........90

3.14. Base and reboiler of Figures 2.14 and 2.15 represented by a DCBase element ...................................................................................................91

3.15. Condenser and accumulator of Figures 2.8, 2.11, and 2.12.........................92

3.16. Hierarchical bond..............................................................................................92

3.17. Hierarchical bond graph of a distillation column ........................................93

4.1. The Gallun distillation column .........................................................................98

4.2. Comparison of the distillate flow of Gallun and simple model in experiment one .......................................................................................106

4.3. Comparison between the stage 35 temperature in Gallun and simple model in experiment one.....................................................................107

4.4. Simple model temperature spread for experiment one ................................108

4.5. Control of base liquid level in simple model in experiment one .................108

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LIST OF FIGURES — Continued

4.6. Control of T35 using steam flow (Wsh) in the simple model in experiment one .......................................................................................109

4.7. Temperature profile of the simple model in experiment two ......................110

4.8. Temperature profile of simple model in experiment three ...........................111

4.9. Reflux flow control in the simple model in experiment four .......................112

4.10. Base liquid height control in experiment four ..............................................112

4.11. Molar fractions of the bottoms product in the simple model with experiment five .....................................................................................113

5.1. A generalized distillation stage.........................................................................119

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LIST OF TABLES

1.1. The basic conjugate pairs ...................................................................................17

1.2. The basic nodes of a bond graph ......................................................................17

2.1. Equations of state ................................................................................................53

2.2. Capacitance of Equation (2-66) .........................................................................54

2.3. Capacitance of Equation (2-67) .........................................................................55

2.4. Capacitance of Equation (2-68) .........................................................................55

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ABSTRACT Modeling and simulating distillation columns is not a new enterprise. All

of the models described in the literature either contain algebraic loops or

simplifying assumptions that render the model ill-equipped for dynamic

simulations. The structure and the equations that represent a trayed distillation

column are explored using bond graphs. Bond graphs model the power flow in

a system, an inherently instructive way to view complex systems. The power of

bond graphs is evident by providing a clear, graphical representation of a

distillation column that systematically organizes the equations and possible

approximations. The model of a distillation column is explored in general and

then by using a specific model developed by Steven Gallun. Results of this

study reveal several ways of eliminating the algebraic loops and producing a

dynamic model. The bond graph model can be expanded by introducing other

elements including chemical reactions and thermal interaction with other

columns.

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1.0 INTRODUCTION

The basic idea of distillation is to separate components of a mixture from

each other to various degrees. A distillation column is one of the primary

techniques used by industry for separating a mixture. A distillation column is a

complex system that is represented by a number of models depending on what

aspects of a distillation column are chosen and on what assumptions are made.

The basic idea of bond graphs is to graphically model a system. As a modeling

methodology, bond graphs represent a powerful approach for understanding

the distillation column. This chapter presents background information for both

distillation columns and bond graphs.

1.1 The basics of a distillation column

The distillation column is a widely used apparatus used to separate

various chemicals, most commonly petroleum products. Historically, distillation

has been around for millennia. Distillation using more than one stage has been

around for a couple of centuries [1]. The theoretical basis of the separation is the

different boiling points of the components being separated.

A simple example should explain this. Assume a two component mixture

in a closed chamber; component A with boiling point at TA and component B

with a higher boiling point at TB. At a temperature between TA and TB, an

equilibrium is reached such that the percentage of component A in the vapor is

higher than the percentage of component A in the liquid. With only two

components, this also means that the percentage of component B in the vapor is

lower than the percentage of component B in the liquid. Now, drain out the

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liquid and a new equilibrium will be reached where the vapor contains an even

higher percentage of component A than in the liquid that will form. If the ratio

of liquid to vapor is to be about the same, the temperature will need to be

lowered. As we continue the process of draining and lowering the temperature,

the result will be a mixture of almost all component A at a temperature just

above TA.

This simple example describes single-stage batch distillation. A

distillation column is a clever way to reproduce many chambers at the same

Figure 1.1. The basic stage of a distillation column.

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time. The chambers in a distillation column, called stages, are stacked on top of

each other separated by a stage plate, or tray. The stage in Figure 1.1 is defined

to contain the foam on the tray, the vapor above the tray and the liquid in the

downcomer below the tray. Thus, Figure 1.1 shows one full stage and two half

stages. Figure 1.2 illustrates a basic distillation column. The liquid is drained by

gravity; the vapor migrates up due to pressure caused by heat. The liquid is

forced to flow over each stage plate before it flows down a downcomer to the

next stage plate. The vapor

is forced to flow through

holes in the stage plate, and

then through the liquid

over the plate, thus creating

good contact between

liquid and vapor.

At the bottom of the

distillation column is a heat

source, the reboiler, and the

ability to siphon off liquid,

the bottoms. The top of the

distillation column is a

condenser that cools the

vapor of the top stage into

liquid and a receiver that

holds the liquid and allows

for some of this liquid to be

Condenser

DistillateAccumulator

Reflux

Reboiler

Bottoms

Feed

Figure 1.2. The basic distillation column.

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returned, the reflux, and some to be taken away as distillate. Somewhere in the

middle of the column is the feed, where the original mixture to be separated is

introduced into the column. Here, vapor rising from below heats the mixture

such that vapor rising to the next stage has a higher percentage of the low-

boiling component (component A) than the original mixture. The liquid flowing

through the downcomer will have a higher percentage of the high-boiling

component (component B). If all has worked well, the distillate will be a nearly

pure component of the mixture, the component with the lowest boiling point.

The liquid flowing down from the top stage, although initially pure, captures

the high-boiling component in the vapor coming up by cooling the vapor.

In a multicomponent distillation, both the distillate and bottoms are

typically composed of several of the original components. The most volatile of

the components will not appear in the bottoms and the least volatile of the

components will not appear in the distillate. The component appearing in both

distillate and bottoms with the highest percentage in the distillate is called the

high key. Similarly, the component appearing in both distillate and bottoms

with highest percentage in the bottoms is the called the low key. Essentially, the

distillation column separates the mixture into the high key and more volatile

components and the low key and less volatile components.

A real distillation process can be complicated by many factors. There

might be many components. The component to be separated out has neither the

highest nor lowest boiling point. The phase equilibria may not be as simple as

presented above (e.g., in an azeotropic mixture, the equilibrium temperature at a

specific ratio of components can be lower than the lowest boiling point of any of

the pure components). The components may form multiple liquid phases. Real

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distillation columns become complex in meeting the requirements. Feeds of

different mixtures can come in on several stages. Distillate and bottoms may be

reintroduced at various stages. Sidestreams can transport liquid or vapor off any

stage to become feeds on another stage or into another column. Heat can be

added or removed from any stage. Several distillation columns can be used in

series (e.g. azeotropic mixtures require more than one column). Heat integrated

columns allow heat flows between columns. Additional components such as

solvents, may be added to facilitate separation. The condenser may not totally

condense the vapor into liquid. A variety of control strategies may be used to

stabilize or optimize the process. Designers of columns must consider these

factors as well as deciding the number of plates, the type of plates (see [1] or [2]

for the choices), whether tray plates are used at all, whether additional

components should be added to facilitate separation, or whether a simple

chemical reaction should occur. The ideas presented in this thesis are hoped to

be basic enough and bond graphing powerful enough to be applied to any of the

possible configurations.

The purpose of a tray is to facilitate liquid and vapor interaction; this

purpose is also fulfilled by a packed column, in which trays are not used at all.

The column is filled, or packed, with irregularly shaped objects designed to

create continuous contact between liquid falling down and vapor rising. This

form of distillation column is gaining popularity in industry [3]. The non-

hydraulic equations are identical with a trayed column, except a stage is no

longer defined by stage plates. The form of the dynamic equations is modeled

intuitively by partial derivatives; approximating this by a series of differential

equations, as if it were a trayed column, makes sense. One notable difference

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between a packed and trayed column is that the pressure differential is much

lower in a packed column; the hydraulic equations for a trayed column could

not be used. The equations for a packed column will not be discussed.

Complex distillation can lead to a variety of problems; some of these

problems must be anticipated when designing or modeling a column. Weeping

occurs when the liquid flows down through the holes in the plate, caused by

insufficient vapor flow [2]. Flooding occurs when the liquid flows up through

the plate, caused either by excessive vapor flow (entrainment flooding) or

excessive liquid buildup on the stage (downcomer flooding) [1, 4]. Entrainment

is either the carrying off of liquid with the vapor through the plate or the

carrying off of the vapor with the liquid through the downcomer. Other

problems include downcomer blocking, oscillations in the liquid holdup, and

instabilities in the operations of the column. These problems can occur in any

distillation column with the right provocation, although design and control

systems should be able to limit the likelihood. Properly modeling and

simulating a column should expose such problems.

1.2 The basics of bond graphs

A bond graph is a graphical description of a physical system that

preserves the computational and topological structure of the system. Each bond

represents the transport of energy as a pair of conjugate variables (effort and

flow) that multiply together to represent power. Thus, bond graphs show the

flow of power through a system. Table 1.1 lists several pairs of conjugate

variables.

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Bonds (half arrows) originate and terminate at various types of elements,

depending on the physical system and the equations used to describe them. The

basic elements describe fundamental relationships between the variables that

are attached by a bond or by bonds. These basic elements are listed in Table 1.2.

As suggested by their definitions, only the 0-node and 1-node accept more than

one bond. These elements will not be sufficient to describe a distillation column;

new elements will be developed to reflect the equational forms of the distillation

effort (e) flow (f)

pressure volume flow temperature entropy flow

force velocity chemical potential molar flow

voltage current torque angular velocity

Table 1.1. The basic conjugate pairs.

symbol name equation 0 0-node ∑

∀i fi = 0

1 1-node ∑ ∀i

ei = 0

R Resistance e = R * f or f = e/R C Capacitance

(Compliance) f = C dedt

I Inductance (Inertia) e = I

dfdt

SF Flow Source f = given SE Effort Source e = given

Table 1.2. The basic nodes of a bond graph.

17

column model and of thermodynamic systems in general.

Bond graphing a system provides more than just a graphical mapping of

the system. Bond graphing inherently provides some system analysis, an easy

means of expanding the system, and modularity. Each bond is assigned one

causality stroke that appears at one end of the bond. The causality stroke

determines which equation (represented by an element) is used to solve for each

variable; the stroke is placed where the flow variable is calculated. Thus, a bond

graph can determine whether a system contains algebraic loops or structural

singularities. Adding an element to the system (e.g., a resistor, a chemical

reaction, a heat source) can be represented by adding an element and a bond to

the bond graph. The effect of such a new element on the system can be seen

immediately by the placement of the causality stroke. Pieces of bond graphs can

be grouped into subsystems and be represented by a new element. Hierarchical

bond graphs are thus modularized to describe large systems.

It is beyond the scope of this thesis to describe bond graphs in all details.

See [5], [6], or [7] for a comprehensive introduction to the subject of bond

graphs and their use. For a comprehensive bibliography of bond graphs, see [8].

Bond graphs have proven to be useful in describing mechanical and electrical

systems. Bond graphs also provide a framework for working with convection

and with thermodynamics [9, 10, 11, 12, 13]. The use of bond graphs in chemical

systems has been limited, however. Chemical systems should benefit from being

bond graphed. Later sections will discuss how bond graphs can be assembled to

represent a distillation column.

1.3 The assumptions of a distillation column

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In forming a model for the distillation column, general assumptions

about the operation of a distillation column must be made. The assumptions a

model makes are the major distinctions between the models found in the

distillation literature. A complete set of dynamic equations would be of a

daunting size considering that a stage contains many forms of energy

transformations and that a distillation column or a set of columns can contain

hundreds of stages. Of course, simplifying assumptions are desirable in such a

complex apparatus. Early computer models required stringent simplifying

assumptions to run [14, 15, 16].

Typical assumptions deal with system constraints (i.e. the type of column

and components used), heat flow, hydraulics, and dynamic equilibrium.

Assumptions leading to the simplest set of equations that approximate the

behavior of a distillation column are used. Assumptions can be broken down

within these categories of assumptions.

System constraints: 1) The condenser may be a partial or total condenser;

2) The trays may be of several types and each requires separate models for

vapor flow through them; 3) Chemical reactions must be modeled if the

components react. Distillation mostly does not include components that react,

though some models do allow for chemical reaction [4, 17, 18]. The most general

models also allow sidestreams, feeds, and heat transfer on each stage, allowing

for a specific complex column to be modeled.

Heat flow: 1) The column is adiabatic, the only heat exchanges by

conduction that need be modeled are those in the condenser and reboiler; 2) The

thermal capacitance of the column metal is negligible. Both of these assumptions

are used by almost all distillation models.

19

Hydraulics: 1) Vapor holdup (of mass and energy) on each stage is

negligible. The molar holdup of the vapor is usually much less than the molar

holdup of the liquid, thus, the vapor holdup effects are considered small. This

also allows for a great simplification in the equations as seen in the next chapter.

Further, more stringent, assumptions can be made that simplify the model

greatly, such as 2) constant pressure or pressures throughout the column, 3)

constant liquid molar holdup, 4) constant liquid volume holdup, 5) constant

reflux flow, or 6) negligible hydraulic dynamics. These simplifying assumptions

are essentially the assumption of small perturbations from steady-state.

Dynamic equilibrium conditions are assumed to be reached between the

vapor and the liquid before leaving the stage, which requires the intensive

variables (i.e. pressure, temperature, chemical potential) of both vapor and

liquid to be equal. In the simple example of single-stage batch distillation,

equilibrium was reached over time before draining and lowering temperatures.

In a distillation column, continuous heat and mass exchange is assumed to

create a dynamic equilibrium. This is the assumption of perfect mixing of vapor

and liquid on the stage. The equations detailing the vapor-liquid equilibrium

(the steady-state flow between the phases resulting in equal chemical potentials)

have a large variety of assumptions that determine their complexity and

whether they form an algebraic loop. The assumption of vapor-liquid

equilibrium is typically relaxed by using Murphree efficiency.

The assumption of negligible hydraulic dynamics is common in the

literature. In making the model as simple as possible, the equations for pressure,

fluid flow, and vapor flow are omitted and values are assumed. Justification of

this comes from Levy, who showed that the (steady-state, algebraic) hydraulic

20

equations had little influence on the most influential modes, the smallest

eigenvalues of a distillation system [19]. The equations dealing with the vapor-

liquid equilibrium were shown to be the most influential. Thus, first or second

order approximations of the distillation column could leave out the hydraulic

dynamics [20]. Work by Tyreus, et. al. and Lagar, et. al. verified that the

eigenvalues most influenced by the (steady-state, algebraic) hydraulic equations

are independent from those due to the vapor-liquid equilibrium, but that

removing the hydraulic equations does not reduce the stiffness of the system

[21, 22]. The assumptions used reflect the aim of the model. If comprehensive

dynamics are to be included, the hydraulic equations must be included.

Assumptions will play a large part in determining the scope and

desirability of a distillation column model. The severe assumptions of steady-

state, neglected hydraulics, or small perturbation models will not be considered

as possibilities. Assumptions and the resultant equations will be dealt with on

an element by element basis in the next chapter.

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2.0 THE ELEMENTS OF A DISTILLATION COLUMN

A complete dynamic model of a distillation column must include

material balance and flow, energy balance and flow, liquid to or from vapor

flow within a stage, temperature, pressure, and hydraulic dynamics, system

constraints, and chemical reactions. Together, these relationships can define the

operations of a distillation column, as well as many other chemical and

thermodynamic processes. For modeling reasons, these relationships can be

seen in terms of power balances. In terms of power flow, the relationships

become temperature and entropy flow, pressure and volume flow, and chemical

potential and molar flow. These relationships and any bond graphs associated

with them are derived as separate elements of a distillation column. This

chapter will discuss the equations that form the traditional model, their origins,

their possible variants, and how they might be reformed into bond graph

notation.

From these equations, three models of a distillation column will be

formed. Equations used in the model of Gallun will be noted with a G. The

equations used in the simple modification of Gallun's model will be noted with

an S. Equations that form a rigorous model that can be bond graphed will be

noted with an R.

2.1 Mass balance

There is an overall balance of mass. This simple statement is carried to

the stage level. The conservation of mass is described by the equation:

22

dMass dt = (rate of liquid coming in – rate of liquid leaving)

+ (rate of vapor coming in – rate of vapor leaving) (2-1)

Many forms of equation (2-1) can be and are developed, depending on the needs

of the rest of the model [4]. Only a few are considered here, with the emphasis

on the mass balance for each stage j. The liquid coming into each stage is mainly

from the downcomer of the previous stage, the vapor coming in is mainly

through the plate from the stage below. Other sources of flow can include the

feed stream which is generally a liquid that comes in on only a few of the stages.

Sidestreams can remove some of the vapor or liquid from a stage in the middle

of the column [1]. Entrainment, both liquid-in-vapor and vapor-in-liquid, can

also be included in the model [1, 2, 23]. This thesis will not consider sidestreams

or entrainment, but results presented here should allow their inclusion.

Consider the mass balance in terms of molar holdup, Mj , and molar flow

for stage j:

dMjdt = (Fj + Lj-1 – Lj) + (Vj+1 – Vj) (2-2)

The liquid component balance for component i uses molar fractions of liquid

(xji ) and vapor (yji ):

d(Mj xji)

dt = (fji + Lj-1 xj-1,i – Lj xji) + (Vj+1 yj+1,i – Vj yji) (2-3)

Notice the assumption implicit in equation (2-3). The vapor molar holdup of

each component is constant, even though the vapor molar fractions are assumed

to vary. The changes in total molar holdup can be attributed solely to the liquid

of the stage. This assumption is not correct, but because the molar content of the

stage is primarily in the liquid, it is often deemed a good assumption. The vapor

23

molar fractions are allowed to vary as functions of liquid molar fractions. This

assumption also allows for the great simplification of obtaining the liquid

holdup without explicitly needing to know the flow from liquid to or from the

vapor within a stage. The variable Vj now contains this piece of information

(LVj = Vj+1 – Vj ). The change in molar holdup of each component as a liquid

on a stage (uji ) becomes [24]:

dujidt = (fji + Lj-1 xj-1,i – Lj xji) + (Vj+1 yj+1,i – Vj yji) (2-4)

GS

xji = uji

∑ ∀i

uji (2-5)GSR

An alternative form of equation (2-4) is derived from equations (2-2) and (2-3)

[25]:

dxjidt =

1Mj

[fji – Fj xji + Lj-1 (xj-1,i – xji) +

] Vj+1 (yj+1,i – xji) – Vj (yji – xji) (2-6) A more rigorous form of equation (2-4) does not assume constant molar

vapor holdup (uvji ):

dujidt +

duvjidt = fji + Lj-1 xj-1,i – Lj xji + Vj+1 yj+1,i – Vj yji (2-7)

R

dujidt = fji + Lj-1 xj-1,i – Lj xji – LVji (2-8)

R

duvji

dt = Vj+1 yj+1,i – Vj yji + LVji (2-9)R

24

In this form, the vapor-liquid equilibrium equations (section 2.4) must provide

not the molar fractions yji , but the actual flow rates of a component from liquid

to vapor, LVji . Very few distillation models consider non-constant vapor

holdup [26].

Mass balance is described in a bond graph as a 0-node. Molar flow and

chemical potential form a conjugate pair. Equation (2-8) is shown in Figure 2.1

as a collection of inflowing and outflowing bonds. The stroke at the end of the

top left bond indicates that the mass-balance is solved for duji/dt at the 0-node.

Solving for the conjugate variable, chemical potential, presents a separate

problem.

2.2 Energy balance

There is an overall balance of energy. The energy balance equation

closely follows the mass balance equation.

d(Energy holdup) dt = rate of energy coming in – rate of energy leaving

0µ

f ji

L x j-1,ij-1

Ljx j

LVji

dujidt

ji

Figure 2.1. Bond graph of liquid mass balance equation (2-8).

25

Except for the condenser and reboiler, the energy coming into or leaving a stage

is solely through convection. This takes into account the assumptions of

negligible thermal capacitance and an adiabatic column. The equation for the

internal energy holdup of a stage (Uj ) is similar to equation (2-7):

dUjdt =

⎝⎜⎜⎛

⎠⎟⎟⎞Ufj +

Lj-1 ULj-1

∑ ∀i

uj-1i –

Lj ULj

∑ ∀i

uji +

⎝⎜⎜⎛

⎠⎟⎟⎞

Vj+1 UV

j+1

∑ ∀i

uvj+1i –

Vj UVj

∑ ∀i

uvji

= dULjdt +

dUVjdt (2-10)

R

Using the assumption of negligible vapor energy holdup, this equation can be

simplified to assign all changes in the stage energy to the liquid.

The thermodynamic relationships for internal energy and enthalpy (H)

state:

dUdt = T

dSdt – P

dVdt + ∑

i=1

c µi

dnidt (2-11)

dHdt =

dUdt + P

dVdt (2-12)

Because the volume of a stage is constant, the change in internal energy of a

stage equals the change in enthalpy. Typically, models for distillation columns

focus on enthalpy; there are approximations of enthalpy that can be calculated

from pressure, temperature, and molar fractions.

One method uses virtual values of the partial molar enthalpies to

estimate the enthalpy of the vapor, and a differential equation for the enthalpy

of the liquid [4, 27, 28]. Ideal gas enthalpies are found through a polynomial

26

expansion of the temperature of known coefficients. The actual enthalpy is

found by adding to this the departure from ideal enthalpy, as predicted by an

equation of state. The virial equation of state is used to calculate the enthalpy

departure function for stage j (Ωj ).

HVji = HIdealji + Ωj (2-13)G

Ωj = Pj ⎝

⎜⎛

⎠⎟⎞ 2 –

1ZZj ⎝⎜

⎜⎛

⎠⎟⎟⎞Tj

BmjTj

– Bmj

( )2 ZZj – 1 (2-14)G

The partial derivative depends on the mixing functions used to determine the

second virial coefficient of the mixture, Bmj . The exact equations used for the

partial derivative (51 of them) can be found in Appendix B and the ‘engder’

macro of Appendix D. The equation for the liquid enthalpy becomes:

dHLj

dt = Hfj +

Lj-1 HL

j-1

∑ ∀i

uj-1i –

Lj HLj

∑ ∀i

uji + ∑

∀i vj+1i H

V j+1i – ∑

∀i vji H

Vji (2-15)GS

Note the assumption that the difference in the vapor enthalpy flow is credited

towards the liquid: the derivative of the enthalpy of the vapor is assumed zero.

Other methods of enthalpy calculation involve various equations of state

[29, 30]. Some models use an algebraic equation for the enthalpy balance [25].

Although this form has been shown to be valid [28], it has also been shown less

accurate [31].

In bond graph notation, energy and mass are automatically balanced:

these equations manifest themselves automatically as 0-nodes and 1-nodes.

Instead of dealing with energy or enthalpy in bulk, the components of

27

thermodynamic energy are considered. The following equations, equation (2-11)

restated for one component in a mixture and one form of the Gibbs-Duhem

equation, reveal the conjugate pairs that bond graphs look at:

dUidt = T

dSidt – P

dVidt + µi

dnidt (2-16)

Gibbs-Duhem: dµidt ni =

dPdt Vi –

dTdt Si (2-17)

The corresponding bond graphs contain two new nodes: Ui is a 3 input

capacitive element and E0 is essentially a 3 input 0-node. The use of these nodes

will be discussed in the next chapter.

2.3 Hydraulic (pressure and volume flow)

Of all the equations within a distillation column model, the hydraulic

equations are the most dependent on the geometry of a particular column, and

thus, the most variable. Various types of vapor holes, multiple-pass trays with

multiple downcomers, circular flow paths, splash baffles and calming zones

before the tray weir, slotted weirs, laminar flow baffles, and various packing

arrangements all require special equations to become modeled.

A comprehensive analysis of stage dynamics would have to include the

partial derivative continuity equation (Navier-Stokes equation) to adequately

U iTdSi /dt dV i /dt

dni/d

t

P

µi

E0dT/dt dP/dt

dµi/d

t

Si V i

ni

Figure 2.2. Internal energy elements.

28

describe the interaction of pressure with fluid velocity, viscosity, surface

tension, and density of the vapor and liquid as they flow through each other.

Bond graphs can be used to describe the continuity equation in fluid flow [32,

33]. For simplicity, flow could be divided into regimes such as spray, froth,

emulsion, bubble, and foam. These regimes have been studied and empirical

functions created relating various hydrodynamic parameters for each regime

[2]. The distillation literature contains many searches for theory and empirical

data that accounts for the formation of foam and spray, the flow patterns that

form upon and effect the hydraulics of a distillation tray, and the prediction of

densities within the different regimes [34]. Yet, such a comprehensive model of

the stage plate would require more information about the components than is

readily available and contains more information than needed. The large number

of untested equations in this type of model suggests more research.

Pressure and volume flow will be determined using Bernoulli’s equation

[35, 36]:

pressure1 – pressure2 = 12 ρ (velocity2

2 – velocity12 )

+ ρ g (height2 – height1) (2-18)

Bernoulli’s equation is essentially balancing the kinetic and potential energy in a

flowing fluid. The assumptions used to derive it are constant density,

temperature, and molar fractions, as well as perfect transfer of energy (no

thermal loss) and steady-state flow. A more complete version of Bernoulli’s

equation balances more forms of energy and uses length as its dimension [37,

38] :

29

⎝⎜⎜⎛

⎠⎟⎟⎞

J U1 + z1 + P1

ρ1 g +

v12

2 g – ⎝⎜⎜⎛

⎠⎟⎟⎞

J U2 + z2 + P2

ρ2 g +

v22

2 g

= ∆hf + ∆h (2-19)

where J is a conversion factor from heat to mechanical energy and ∆h is the

external work added to the flow. The term ∆hf is the pressure drop (called head

loss when length is the dimension) due to friction. The head loss can be related

to velocity of foam (uf), length of flow path (lf), hydraulic radius (rh), and a

friction factor (f) [27]:

∆hf = f uf

2 lf 12 g rh

(2-20)

The friction factor is related to the Reynolds number, and is therefore viscosity

based. In most instances, this factor is quite small compared to the kinetic and

potential energy terms.

Where models in the distillation literature have sought to include the

liquid flow through the downcomer and vapor flow through the stage,

Bernoulli’s equation is used. The nature of the derivation of Bernoulli’s equation

is evident by examining a more comprehensive equation of motion with mass,

velocity and forces.

d(m v)

dt = ∑F (2-21)

The equation as it stands is suited for a Lagrangian frame of moving solids. The

equation needs reforming to be used in a Eulerian fixed frame of flowing fluid,

pressure differences and volume flows. This is achieved by dividing each side of

30

equation (2-21) by area and recasting mass and velocity into density (ρ), volume

(V), cross-sectional area (A), and volume flow(Q):

1

A0

ddt⎝⎜⎛

⎠⎟⎞ ρ V Q

A = ∑P (2-22)

Assume equation (2-22) refers to a section of pipe with constant a volume and a

fluid with a constant density; only the volume flow and area are left within the

derivative. The sum of pressures in this equation is the sum of the pressures at

the input and output, the pressure caused by the fluid moving in a gravitational

field (potential energy), and any frictional loss pressures. Thus, equation (2-22)

becomes

ρ V

A0 A2 (Q

• A – Q A• ) = P1 – P2 + ρ g (height1 – height2) – Pf (2-23)

The term A0 denotes the average cross-sectional area. When the derivative of

volume flow is zero, steady-state flow requires the pressure differential to be

constant, hence a constant average acceleration through the tube. Part of this

pressure differential can be caused by a change in the cross-sectional area of the

tube; this is the kinetic energy term of Bernoulli’s equation. The derivative of

area in equation (2-23) is better formed using the derivative of velocity,

acceleration.

Q = v A

Q• = v• A + v A• = 0

A• = – Av v

• = – A2

Q a (2-24)

The kinetic energy term is thus calculated as

31

– ρ V Q A•

A0 A2 = ρ ∆x a = PKE (2-25)

ddt PKE = ρ v a (2-25a)

where a is the average acceleration through the tube. Equation (2-25a) only

makes sense in the Lagrangian sense of a changing ∆x; in the Eulerian sense, the

derivative is zero. The actual tube may not be shaped to give a constant

acceleration through the tube, but the average acceleration will always be the

same as if the acceleration is constant. Acceleration is derived using the

traditional method of deriving kinetic energy [35]. The main assumption is

movement with constant acceleration.

a = constant acceleration

v2 = v1 + a t t = 1 a (v2 – v1)

x2 = x1 + v1t + 12 a t

2 = x1 + 1 a v1 (v2 – v1) +

1 2 a (v2 – v1)

2

a = v2

2 – v12

2 ∆x = Q2

2 ∆x ⎝⎜⎛

⎠⎟⎞1

A 22 –

1A1

2 (2-26)

As long as the volume flow is constant, the average acceleration of flow through

the tube will equal this constant acceleration. Combining equations (2-23), (2-

24), (2-26), a full expression of fluid flow in a tube is realized.

ρ V A0

2 Q• +

ρ Q22

⎝⎜⎛

⎠⎟⎞1

A 22 –

1A1

2

= P1 – P2 + ρ g (height1 – height2) – Pf (2-27)

Note that the kinetic energy term is of energy stored, not of energy lost. A true

energy loss term would change sign with a change in the direction of flow. The

second term in equation (2-27) is represented by a KE-element in Figure 2.3;

32

although it is essentially a resistive element as long as flow remains in one

direction, it should not be represented by an R-element [39, 40, 41]. The

potential energy term is also an expression of energy stored. The form of

equation (2-27) is that of an inertia with various other elements attached:

∆P + PKE = P1 – P2 + PPE – Pf (2-28)

∆P = I ddt Q =

ρ V A0

2 Q• (2-28a)

PKE = ρ ∆x a = ρ Q2

2 ⎝⎜⎛

⎠⎟⎞1

A 22 –

1A1

2 (2-28b)

PPE = ρ g (height1 – height2) (2-28c)

Thus, a bond graph of this equation, using volume flow as the flow variable and

pressure as the effort variable, would center around a 1-node as in Figure 2.3.

Note that the potential energy term is not modeled using an energy storage

element; any tube used is not likely to be so long as to experience a change in

gravitational acceleration. At steady-state, the volume flow and pressures

become constant, which converts equation (2-27) into Bernoulli’s equation.

1

I KE

P1 P2Q

PKE

PPE

²P

SE R

Pf

Figure 2.3. Bond graph of fluid flow, equation (2-28).

33

Although constant flow is a common assumption, the fluid flows in a distillation

column are not always subject to constant pressures. Also, this assumption will

lead to an algebraic loop in the vapor flow, depending on how pressure is

calculated.

The fluid flow within a distillation column can almost all be modeled

using fluid flow in a pipe, equation (2-23). The major difference is that within a

distillation column, the flows are not always subject to the constant volume

assumptions of tube flow. Hydraulic power flow is considered as the conjugate

pair of pressure and volume flow. Referring to Figure 2.4, the hydraulic flow

variables are the vapor flow from stage j (QVj ), the liquid flow over the weir

Figure 2.4 The hydraulic variables in a distillation column stage.

34

(Qsj ), and the liquid flow leaving the downcomer (QLj ). The three areas of

interest are the foam flow over the weir, the liquid flow through and out of the

downcomer, and the vapor flow through the sieve plate.

Consider first the flow through the downcomer. The interaction between

liquid and vapor is assumed to have stopped, the foam coming over the weir is

quickly reduced to clear liquid. The constriction at the bottom, under the

downcomer, limits the flow. The flow is determined by the height difference

between the height of liquid in the downcomer (Zj ) and the height of liquid on

the next stage plate (hL j+1 , derived later), as well as the pressure differences

between the stages. All of these quantities vary dynamically. The height of the

liquid in the downcomer is dependent on the flows into and out of the

downcomer:

dZjdt =

1AD

dVdcdt =

Qsj – QLjAD (2-29)

SR

The area of the downcomer (AD) is much larger than the area under the

downcomer (AUD) through which the liquid flows to the next stage; thus, the

velocity of flow is only considered under the downcomer. All but the exiting

flow is within the area of the downcomer; thus, the average area is essentially

AD. Equation (2-22) can be rephrased in terms of the flow through the

downcomer,

ρLj A0

ddt⎝⎜⎛

⎠⎟⎞Vdc QLj

A = Pj – Pj+1 + ρLj g ⎝⎛ ⎠⎞Zj – h

L j+1 – Pf (2-30)

and combined with equation (2-29):

35

ρA0

2 ⎝⎜⎛

⎠⎟⎞dQLj

dt Vdc + (Qsj – QLj) QLj + ρ QLj

2

2 ⎝⎜⎛

⎠⎟⎞1

AUDj2 –

1ADj

2

~= ρ ZjADj

dQLj

dt + ρ

ADj2 (Qsj – QLj) QLj +

ρ QLj2

2 AUDj2

= Pj – Pj+1 + ρ g (Zj – hL j+1 ) (2-31)SR

∆P = ρ ZjADj

dQLj

dt (2-31a)

PPE = ρ g (Zj – hL j+1 ) (2-31b)

Pdc = – ρ

ADj2 (Qsj – QLj) QLj + Pj (2-31c)

PKE = ρ QLj

2

2 AUDj2 (2-31d)

The kinetic energy term, equation (2-31d), should be immediately compensated

for (eliminated) by the expansion of the area through which the liquid flows

onto the next stage plate. Instead, the kinetic energy term is kept as an estimate

of resistance, as equation (2-20) for resistive head loss does not pertain here. The

DC

I R

Pj Pj+1Qsj

PKE

PPE

²P

SE

PdcjQLj 1

Zj

QLj

Figure 2.5 Bond graph of downcomer hydraulics, equation (2-31).

36

bond graph of this equation, Figure 2.5, is quite similar to the bond graph of the

hydraulics of a tube, except the change in downcomer volume (dVdc) depends

on both the differences in volume flows and pressures. A new element is created

to implement this term. The thin arrowed line indicates that the DC element is

used to calculate Zj , equation (2-29), for the potential energy term.

Assuming that the potential energy considerations are the predominant

forces on the flow, essentially assuming constant flows, Bernoulli’s equation can

be used and the volume flow simplifies to

QLj = Cx AUD 2 g ⎝⎜⎜⎛

⎠⎟⎟⎞

Pj – Pj+1 g ρLj

– h Lj+1 + Zj (2-32)G

Note the inclusion of Cx, accounting for vena contracta effects. Flow through the

downcomer is often considered to have negligible dynamics and is omitted from

many models. The bond graph of equation (3-32) is similar to Figure 2.5 without

the I element or DC element.

For the flow over the weir, the most common equation used is Francis’s

weir formula, relating the height of foam over the weir (hfowj ) to foam volume

flow (Qfsj ), weir length (Lw), and a vena contracta factor (Cd) [2]:

hfowj = ⎝⎜⎛

⎠⎟⎞

3

2 Cd 2 g Qfsj Lw

23 (2-33)

The equation assumes the tray and the components are such that the full height

of foam on the tray is intact as the fluid flows over the weir (i.e., there is no

calming zone). Using relative froth density (φ), clear liquid flow (Qsj ) can be

solved for:

37

φ = ρLjρfj

= hLjhfj

= QsjQfsj

= howjhfowj

Qsj = 2 3 Cd Lw

2 g φ ( )howj

1.5 (2-34)

One modification found in the literature is to use a weir constriction factor (Fw)

and a foam factor (Foamj) and solve for the clear liquid height over the weir [1,

27]:

Foamj = hLj

hwj + howj =

φ + 12 = f(velocity of vapor)

howj = 1.426

g1/3 Fw

⎝⎜⎛

⎠⎟⎞

QsjLw

23 (2-35)GSR

The weir constriction factor modifies the effective weir length much the same as

Cd corrects for vena contracta effects. Francis’s weir equation or its modification

serves as the minimum hydraulics for some models [23, 42]. It can be derived

from Bernoulli’s equation [2]. Using equation (2-18), choose point 1 to be the top

of the foam in the middle of the column and point 2 to be a distance y down

from the top of the foam and just over the weir. The pressures are both equal to

the stage pressure. Assume the velocity at point 1 is much less than at point 2

and solve for velocity at point 2:

v2 = 2 g y

The velocity is only a function of depth. Note that at a depth of zero, there is no

liquid flowing over the weir. The differential volume flow becomes:

dQsj = dA v = Cd Lw dy 2 g y

38

which can be integrated from zero to hfowj to form equation (2-33). The

assumption of constant pressure acting on the foam flowing over the weir is

probably justified, unless the weir is short enough (as in a column of small

diameter) that the flow forms a seal above the downcomer.

In most cases, Francis’s formula is worth using. The only quantity to vary

dynamically is the height over the weir. The derivation of a dynamic equation

for the flow over the weir is similar to that of the flow through the downcomer.

Starting with equation (2-30), and using pertinent information for a tray,

ρLj ANφ A0

2 ⎝⎜⎛

⎠⎟⎞

hfj dQfsj

dt + Qfsj dhfowj

dt + ρLj Qfs j

2

2 φ [ ]Cd (hfowj + Lw) 2

= Pj – Pj+1 + ρ g y – Pf (2-36)

dQfsjdt =

1φ ⎝⎜⎛

⎠⎟⎞dQsj

dt – Qfsj dφ dt

The derivative of the height over the weir is not just the difference in volume

flows into and out of the tray, but must include the vapor to liquid flow and the

feed, as well as the dynamic relative froth density. The height over the weir can

be found rather simply as:

hfowj = hLjφ – hwj =

∑ ∀i

uji

_ρLj

– AD Zj

AN φ – hwj

39

howj = hLj

Foamj – hwj =

∑ ∀i

uji

_ρLj

– AD Zj

AN Foamj – hwj (2-37)GSR

The derivative of the height over the weir is not as simple to find. Francis’s weir

formula will be assumed for use by the models developed.

In terms of bond graphs, the height over the weir represents the volume

of fluid on the tray plate. Rearranging equation (2-37) reveals the equation for

the tray volume.

Vtray = (howj + hwj ) AN =

∑ ∀i

uji

_ρLj

– AD Zj

Foamj

Then the flow over the weir is a function of the tray volume. Equation (2-34) can

be then represented as in Figure 2.6a, by an effort source of gravity and

resistance of kinetic energy. What is not fully conveyed by this graph is that the

FwPj PjQsj

1Pj PjQsj

SEPPE

PKE

R:KE

howj

Qsj

Figures 2.6a and 2.6b. Bond graphs of Francis weir formula, equation (2-34).

40

pressures of both these elements must be equal. The Fw element, shown in

Figure 2.6b, encompasses the effort source and the resistance.

Vapor flow through the plate depends on the type of plate. The equations

for a sieve tray are developed here. Equations for steady-state flow for bubble

cap and valve trays exist [1]. For a sieve plate, the vapor flow is through small

holes in the stage plate and is considered incompressible. The area of the holes is

significantly smaller than the total area of the plate. The vapor between the stage

below and the plate is considered to be well mixed and moving at a much lower

velocity. Using equation (2-27) as a starting point, the volume being considered

is the volume within the holes of the sieve plate, Vhj .

ρ Vj+1 Vhj(Cvc AHj)

2 dQVj+1

dt + ρ Vj+1 QVj+1

2

2 (Cvc AHj)2

= Pj+1 – (Pj + g ρLj h

Lj ) – g ρ

Vj+1 LHj – Pf (2-38)

Vhj = AHj LHj

The stage plate is assumed thick enough that vena contracta effects (Cvc) are

determined within the holes. Rearranging for the derivative, and assuming the

vapor potential energy term to be negligible (i.e. LHj

41

Pf = 4 σDh (2-40)

The assumption of constant forces allows Bernoulli’s equation to be used for

vapor flow from a stage. Also assuming that the frictional forces are negligible,

the vapor flow becomes,

QVj+1 = Cvc AHj 2

ρ Vj+1 ⎝⎛ ⎠⎞Pj+1 – Pj – g ρ

Lj h

Lj (2-41)G

The distillation models in the literature that include an equation for vapor flow

use equation (2-41). The bond graph of vapor flow, equation (2-39), is essentially

that of Figure 2.3.

The assumption behind the derivations of these hydraulic equations is

that pressure is being specified somewhere. An equation of state can be used for

pressure. Unfortunately, the equation of state is usually used to solve for other

variables, such as the molar density or compressibility factor. This, of course,

leads to algebraic loops. A full hydraulic bond graph of a distillation stage will

be developed in Section 3.1.

2.4 Vapor-liquid equilibrium (chemical potential and molar flow)

The equations related to vapor-liquid equilibrium are meant to predict

the flow of a component within a stage from vapor to liquid. The relationship

between the molar fraction in liquid (xi) and molar fraction in vapor (yi) is

generally not as simple as described in the example explaining distillation

(Section 1.1). Depending on the components and assumptions used, these

equations can form a large part of the model. This relationship is the most

important within the distillation column model. Levy, et al. showed that the

42

largest time-constants of a distillation model were due to composition dynamics

[19]. Ranzi, et al., found that up to 35% of the time in a distillation simulation

was taken by vapor-liquid equilibrium calculations [31]. The essential problem

of determining this flow is that chemical theory on this point is inadequate:

there are no equations for chemical resistance or chemical capacitance to relate

chemical potential and molar flow between the liquid and the vapor phases.

Instead of calculating the flow, the traditional method of determining

molar content is to assume that after good contact, vapor and liquid reach

equilibrium conditions. If equilibrium is assumed within a closed system with a

vapor phase and a liquid phase, there are no chemical reactions and pressure,

temperature, and especially the total Gibbs free energy are held constant:

dGdt = ∑

i=1

c

µLi dnLidt + ∑

i=1

c

µVi dnVi

dt = 0

In a closed system without reaction, it must also be true that molar flows are

equal and opposite:

dnLidt = –

dnVidt (2-42)

Thus, the chemical potentials of a component in both liquid and vapor are equal:

µLi = µVi (2-43)

This is the fundamental relationship used to predict multicomponent phase

equilibria [37].

Chemical potential is not used in the literature. What follows from

equation (2-42) is that at equilibrium, the fugacity of a component in a liquid

43

equals the fugacity of that component in vapor [4]. Fugacity of a component in a

mixture, f̂i , is defined for a constant temperature:

dµi = R T d⎝⎛ ⎠⎞ln(f̂i) (2-44a)

limP∅0 f̂i

xi P = 1 (2-44b)

If the mixture is a vapor, in an ideal mixture, and the ideal gas law holds, then

fugacity becomes the partial pressure of the component. This is evident from the

Gibbs-Duhem equation (2-17) combined with ideal mixture (the use of partial

pressure), constant temperature, and the ideal gas law.

(Gibbs-Duhem) dµidt ni =

dPidt V (2-45)

(Ideal Gas Law) V = ni R T

Pi

(Fugacity) dµidt =

R T Pi

dPidt = R T

d dt ln(Pi)

Essentially, the fugacity of a component is a measure of the partial pressure’s

deviation from Gibbs-Duhem and Dalton’s law (ideal mixture). Partial pressure

itself is still used in the traditional model, unqualified by deviations.

Measurement of the actual system pressure does not reveal fugacity, but

pressure. Chemical potential, which is not used in the traditional model, is

essentially absorbed into fugacity along with any deviations a component may

exhibit. In bond graph terminology, these equations suggest a transformer using

molar density as the transformation factor.

ni = _ρi V (2-46)

44

dµidt =

_ρi

dPidt =

_ρi

df̂idt (2-47)

There are several ways to compute and approximate fugacity, depending

on the components used, each involving molar fractions. Thus, a relationship

between xi and yi is found by equating vapor and liquid fugacities. The general

equilibrium equation for this relationship can be stated as:

yi = Ki xi (2-48)GS

where Ki (referred to as the distribution coefficient [30] or K-value) can be a

simple or a complex function, depending on what approximations for fugacity

are used. The most basic expression for Ki, henceforth referred to as the ideal K-

value, is

KIi = PSiP (2-49)

S

where PSi is the vapor pressure (saturation pressure) of component i (a function

of temperature only), and P is the total pressure of the system. The ideal K-value

is the same as Raoult’s law:

P yi = PSi xi

This equation makes sense; for a pure substance, the equilibrium pressure for

two phases is, by definition, the vapor pressure. Thus, in a mixture, the partial

pressures of both liquid and vapor should be equal. If the vapor pressure of a

substance is twice the system pressure, then the vapor will have twice the molar

concentration of the liquid. The implication of this equation is that partial

pressure, not chemical potential, is driving molar flow between the phases. As

TMρ⎯

dµi/dtni

dPi/dtV

Figure 2.7 Transformer representing chemical to hydraulic power conversion.

45

may be expected, in mixtures, the partial pressures are not uniformly equal for

all substances and may serve as a poor model for molar flow. Hence, fugacity

has been introduced to replace partial pressures. Nevertheless, the essence of

these equations is based on the idea that a difference in partial pressure

(fugacity) is driving molar flow between the phases.

For a more detailed expansion, fugacity can be defined in terms of a

fugacity coefficient ϕ̂i (measuring the deviation from partial pressure) , ideal

fugacity coefficient ϕi (measuring the deviation from pressure as a pure

component), activity coefficient γi (measuring the deviation from Dalton’s law),

ideal fugacity fi, ideal vapor pressure, and pressure [1, 4]:

f̂ Li = γLi xi f

Li (2-50a)R

f̂ Vi = ϕ̂V i yi P (2-50b)R

f Li = PSi ϕLi (2-50c)R

If the substance forms an ideal gas and an ideal mixture and conforms to Gibbs-

Duhem, then the fugacity and activity coefficients are one and Raoult’s law can

be used. Combining these definitions, the relation between the molar fraction of

a component in vapor and in liquid can be calculated several ways:

yi = γLi PSi ϕ

Li

ϕ̂Vi P xi (2-51)

yi = γLi K

Ii ϕ

Li

ϕ̂Vi xi (2-52)G

Equations (2-51) and (2-52) are but the start of many various ways in which

molar fractions are calculated. Estimating fugacity and its coefficients occupies a

substantial segment of the chemical engineering literature. Some methods of

46

predicting

Error!

The method used by Gallun will serve as proper illustration [24]. Using

equation (2-51), we must come up with equations for PSi, ϕLi ,

ϕ̂Vi , and γLi . The Antoine equation is used to calculate PSi:

log( PSi ) = Ai + Bi

T + Ci (2-53)GSR

where T is the only variable. Gallun’s equations used for ϕLi were taken from a

1967 monograph by Prausnitz, et al.

ϕLi = ϕSi exp⎝

⎜⎛

⎠⎟⎞

– αLi PSi

R T (2-54)GSR

ϕSi = exp( f1(TR) +ωi f2(TR) ) (2-55)GSR

Temperature is the only variable. To calculate

Error!

ZZ = 1 + Bm _ρ =

1 + 1 + 4 Bm P

R T2 =

P Vn R T (2-56)

GR

Bm is the second virial coefficient of the mixture, which must be calculated using

the following mixing rules:

Bm = ∑i=1

c ∑ k=1

c yi yk Bik (2-57a)GR

Bik = R TCCik ( f3(TR) + ωHik f4(TR) + f4(µR,TR) + ηik f6(TR) )

PCCik (2-57b)GR

TR = T

TCCik (2-57c)GR

47

where the symbols here are constants or functions of yi and T [24]. Then

ϕ̂Vi can be calculated using the following:

ln⎝⎛

⎠⎞ϕ̂Vi = 2

_ρV ∑

k=1

c yk Bik – ln(ZZ)(2-58)GR

Gallun uses the Wilson equation to calculate γLi . Again, the only system

variables that are present are molar fraction (xk) and temperature (T).

ln(γLi ) = 1 – ln⎝⎜⎜⎛

⎠⎟⎟⎞

∑ m=1

c xm Λim – ∑

k=1

c

xk Λki

∑ m=1

c xm Λkm

(2-59a)GR

Λik = αLkαLi

exp⎝⎜⎛

⎠⎟⎞

– (λik – λii)

R T (2-59b)GR

αLi = 1

_ρLj

= f(T) (2-59c)GSR

Expanding these equations for all components for every stage, it is easy to

understand how these might form a significant portion of a model.

The problem of this method and all methods presented in the above

references are that equations for γLi involve xi and equations for

Error!

Another problem with using vapor-liquid equilibrium equations to find

the vapor molar fraction (yi ), is too many equations. Equation (2-48) and the

basic definition of molar fractions:

1 = ∑∀i

yi

48

form one more equation than is needed to find the values of the fractions. The

system is over-determined and generally is resolved by iterating the value of the

temperature until all equations are satisfied. A variation of this method is

presented in Section 2.5.

Currently, the only modification to equating liquid and vapor fugacities

is the use of an efficiency term. Several definitions have been proposed and

developed [2, 46, 47, 48, 49]. The vaporization and modified Murphree plate

efficiencies are defined as [4]:

Eji = f̂ Vi

f̂ Li =

yjiKji xji

(2-60)

EMji = yji – yj+1i

Kji xji – yj+1i =

Eji – yj+1iKji xji

1 – yj+1iKji xji

(2-61)

Ideally, the efficiency is one: the vapor and liquid fugacities are equal. Values

for modified Murphree efficiency are typically found empirically through some

estimate of plate dynamics, but can be derived using film theory, phase

resistances, and transfer units for steady-state flows [4]. One theoretical basis for

efficiencies is that chemical potential differences are driving molar flows

between phases. Using the definition of fugacity, chemical potential differences

are modified to become fugacity differences. Efficiencies are mostly used in

designing a column to discriminate between the number of ideal trays and

number of real trays that are needed.

49

2.5 Heat (temperature and entropy)

The conjugate pair for heat is entropy flow and temperature. Entropy is

not used in traditional models for distillation columns, there can be no bond

graph of these equations. Although there are tables of entropy values to be

found, they are not extensive enough in terms of substances covered, to be

useful. Instead, temperature is found through the over-determined set of vapor-

liquid equilibrium equations, described in Section 2.4. Otherwise, the dynamic

equation for temperature is typically derived from some function of

temperature and time, f(Tj,t):

ddt f(Tj,t) = Tj

f(Tj,t) dTjdt (2-62)

S

An appropriate function depends on the model. Because convection is the major

source of heat flow, the function should incorporate molar flow.

The dynamic equation for temperature can be derived from the vapor-

liquid equilibrium equation. Begin with the basic equations, sum over all i, and

then take the derivative:

yji = Kji xji

1 = ∑ i=1

c yji = ∑

i=1

c Kji xji

0 = ∑ i=1

c ⎝⎜⎛

⎠⎟⎞dKji

dt xji + Kji dxjidt (2-63)

If the ideal K-value is now assumed and the Antoine equation used, then K is a

function of Pj and Tj. The derivative of K can be stated:

50

dKjidt =

KjiTj

dTjdt +

KjiPj

dPjdt (2-64a)

KjiTj

= Bi Kji ( )Tj – Ci -2 ln(10) (2-64b)

Now, let the derivative of Pj be zero, combine equations (2-63) and (2-64), and

solve for the derivative of Tj:

dTjdt = –

∑ i=1

c

Kji dxjidt

∑ i=1

c Bi Kji ( )Tj – Ci -2 ln(10) xji

Another function used is one for enthalpy, an approximation for enthalpy

in liquid as a polynomial function of temperature.

HLji = ai + bi Tj + ci T2j (2-65)GS

Because the liquid enthalpy is calculated using a differential equation (2-15), the

derivative of Tj can be found similarly to the derivation above, without having

to assume a constant pressure. Again, the key here is that molar flow is the

assumed major conveyer of entropy.

Entropy is also flowing from the frictional loss terms of the hydraulic

equations, but this is not considered to be a significant source of heat. The real

heat flow by conduction occurs in the reboiler and condenser, which will be

dealt with in Sections 2.7 and 2.8.

2.6 Equation of state

51

An equation relating the steady-state pressure, temperature, volume and

number of moles is an equation of state. The equation can serve as a source of

behavior of a system. Usually, an equation of state is used to solve for the

system pressure of a stage. If the vapor molar holdup is assumed constant,

however, the molar volume (molar density) must be calculated from the

equation of state. Such a model usually uses Bernoulli’s equations to calculate

pressure and is forced to either assume values for liquid flow or use iterative

methods. The assumption of dynamic equilibrium means the pressure of liquid

and vapor are equal.

The equations of state considered are the ideal gas equation, the virial

equation of state, and the Peng-Robinson equation of state. Of these equations,

only the last one is valid for both liquid and vapor phases. Peng-Robinson does

assume that molar volume is known, it can not predict what percentage of the

substance is in which phase. Hundreds of equations of state exist [43]; these are

considered only as common examples. Also considered is a fluid that does not

change volume with a changing pressure, the incompressible fluid. Table 2.1

lists these equations of state. Which equation of state should be used will

Ideal P = RTv , v =

Vini

Virial P = R T Z

v , Z = 1 + Bv

Peng-Robinson P = RT

(v – b) – α a

v(v + b) + b(v – b)

α , a, b are functions of temperature, critical pressure and temperature, and the acentric factor

Incompressible v = a + b T + c T2

Table 2.1 Equations of state.

52

depend on the nature of the distillation components, as well as the information

available to compute constants within an equation of state. For instance, the

second virial coefficient (B) is detailed in equations (2-57).

Several capacitive relationships can be set up using these equations of

state, the internal energy equation and the Gibbs-Duhem equation. The

relationships are:

dSidt = Cs

dTdt (2-66)

Equation of State Cs = dSidt /

dTdt

Assuming

(All assume dUidt = 0)

Ideal µi niT2

dPdt =

dVidt = 0

PT

dPdt =

dnidt = 0

Virial µi niT2

dZdt =

dPdt =

dVidt = 0

PT

dZdt =

dPdt =

dnidt = 0

Peng-Robinson µi RT(v – b)

niP

dPdt =

dVidt = 0

R v PVi(v – b)

niP

dPdt =

dnidt = 0

Pni

= v R T

(v – b)2 ni –

2 α a v (v + b)ni ( )v + ( 2 + 1) b 2 ( )v – ( 2 – 1) b 2

Incompressible – µi niVi T (b + 2 c T)

dPdt =

dVidt = 0

P niVi (b + 2 c T)

dPdt =

dnidt = 0

Table 2.2 Capacitance of Equation (2-66).

53

dVidt = Cv

dPdt (2-67)

dnidt = Cn

dµidt (2-68)

The relationships are only true under certain conditions. The values of Cs, Cv,

and Cn are shown in Tables 2.2, 2.3 and 2.4 along with the required conditions.

Note that none of these conditions are true in a dynamic distillation column. As

the equation of state is usually used for pressure calculation, equation (2-67) is

usually represented in a bond graph as a C element.

Equation of State Cv = dVidt /

dPdt Assuming

Ideal / Virial Vi2

ni R T dTdt =

dnidt = 0

Vi2 ⎝⎜⎛

⎠⎟⎞

1 – Si TVi P

Si T

dµidt =

dnidt = 0

Peng-Robinson v Vi R ⎝⎜⎛

⎠⎟⎞

1 – Si (v – b)

Vi RSi (v – b)

dµidt =

dnidt = 0

Incompressible Vi ni (b + 2 c T)

Si dµidt = 0

Table 2.3 Capacitance of Equation (2-67).

54

The value of Cn for the ideal gas equation is that used by Oster, et al. [50].

In describing the molar flow between phases, only the one for Peng-Robinson is

adequate in describing both phases. These relationships can only be used in

conjunction with equations that relate chemical potential to molar flow.

Unfortunately, as described in Section 2.4, no equations of state nor any other

equations exist to provide such a relationship. For instance, equation (2-68)

could be used to solve for chemical potential of both the liquid and the vapor in

a stage, based on the influx of total moles. The equation can not tell us the flow

between the phases because it is only valid for constant volumes. Nor would it

make sense to use such a storage element to describe phase flow.

2.7 Condenser and receiver

The condenser and the receiver are separate from the distillation column,

though essential for the distillation process. The condenser accepts the vapor

rising from the top stage plate and removes a significant amount of heat. The

main choice is between total and partial condensers, depending on the type of

modeling simplification desired. A total condenser is assumed to remove

Equation of State Cn = dnidt /

dµidt

Assuming

Ideal ni

R T dVidt =

dTdt = 0

Virial ni

R T Z dZdt =

dVidt =

dTdt = 0

Peng-Robinson niVi

niP

dVidt =

dTdt = 0

Incompressible 1 dUidt =

dVidt =

dTdt = 0

Table 2.4 Capacitance of Equation (2-68).

55

enough heat to totally condense the vapor, that amount of heat varying with the

condition of the vapor flowing in. A partial condenser is assumed to remove a

fixed amount of heat, the amount of condensation varying with the condition of

the vapor flowing in. The thermal equations and the hydraulic equations

developed will be for a total condenser. No change in chemical potential is

assumed; phase changes are being driven by entropy flow.

The thermal aspect of the model is power flow due to convection,

conduction and heat of vaporization. The vapor is brought up against tubes

filled with cooling water. The amount of heat removed is controlled by the flow

of cooling water, which in turn is controlled by a valve. Conduction is a function

of thermal capacitance and thermal resistance. The thermal equations are of the

form,

heat flow = (volume flow) (heat capacity) (∆Temperature)

= (heat transfer film coefficient) (area of heat transfer) ∆T

change in heat holdup = (mass) (specific heat capacity) dTdt

The mass of the flowing cooling water, the mass of the vapor rising from the top

plate, and the mass of the condenser tubes must be considered in these

equations. Because total condensation is assumed, the desired variable is the

heat flow from vapor (QC). In the bond graph terminology of entropy flow, the

above forms can be converted [5].

entropy flow = (volume flow) (heat capacity) ∆T / T

= (heat transfer film coefficient) (area of heat transfer) ∆T / T

change in entropy holdup = (mass) (specific heat capacity) •T / T

56

Specifically, for the condenser described in the dissertation of Gallun [24], the

original heat flow equations are:

Ucw Ccw •T wo = Qcw Ccw (Twi - Two) + Uwm Aw (Tmc – Two)

(2-69a)GS

Umc Cmc •T mc = Uwm Aw (Tmc – Two) – QC (2-69b)GS

QC3 = [QV2]REF

QV2 Upmo Apc (Tj – Tmc)3 (2-69c)GS

The subscripts refer to cooling water (cw), water out (wo), water in (wi), and the

mass of the cooling tubes (mc). The equation for the heat flow from the vapor,

equation (2-69c), is an empirical approximation from reference values of vapor

and heat flow. Tj in this equation is considered to be some average temperature

between T1 (receiver temperature) and T2 (stage 2 temperature). Unlike pure

substances, the distillation mixture will not condense at just one temperature. In

terms of entropy flow, these equations become equations (2-70).

•S wi + ∆•S 1 = •S mc + •S wo (2-70a)R

•S mc + ∆•S 2 = QC/Tj + •S z1 (2-70b)R

•S wi = Qcw Ccw (Two – Twi) / Two (2-70c)R

∆•S 1 = Ucw Ccw •T wo / Two (2-70d)R

•S mc = Uwm Aw (Tmc – Two) / Two (2-70e)R

∆•S 2 = Umc Cmc •T mc / Tmc (2-70f)R

•S wo = Uwm Aw (Tmc – Two)2 / Two2 (2-70g)R

QC/Tj = Upmo Apc (Tj – Tmc)/Tj (2-70h)R

•S z1 = Upmo Apc (Tj – Tmc)2/(Tmc Tj) (2-70i)R

57

The equation for •S wo reflects the idea that the power lost in the resistance

between the cooling water and the condenser tubes is regained as heat flow into

the cooling water. The assumption that (Two >> (Tmc – Two)) effectively

eliminates this term from Gallun’s equations (2-69). Figure 2.8 shows the bond

graph associated with equations (2-70). The R element, equation (2-70a), in

contrast to the RS element, shows that power is lost to convection as the cooling

water flows away, and is modulated by the volume flow of cooling water, Qcw.

Knowing the heat loss (QC), the enthalpy-balance for the receiver becomes a

variation of equation (2-15):

dHL1

dt = – L1 H

L1

∑ ∀i

u1i + ∑

∀i v2i H

V 1+1i – QC (2-71)GS

The hydraulic aspect of the condenser is the volume flow of vapor

entering (QV) and the liquid flow leaving. The mass holdup in the condenser is

assumed to be constant and at temperatures that vary between T1 and T2. The

model assumes a weighted average of T1 and T2. The volume flow of vapor is

QC/T j10SE 1 0

RSR

C:cooling water

C: cooling tubes

Twi Two Two Tmc Tmc

²T Swi•

²S ²S

Swo

SmcSmc QC/T j

²T

Swi•

Swi•

Two Tmc

•

•

•

• •

RS

Sz1 ²T•

1Tj

Figure 2.8. The bond graph of the condenser, equations (2-70).

58

calculated using an empirical estimation based on pressure differences and the

vapor’s molar density. Also to be considered is the volume flow of cooling water

through the cooling tubes, Qcw. The pressures at the inlet and the outlet of these

tubes are considered constant and the flow is controlled by means of a valve.

Figure 2.9 reveals this simple system.

The hydraulic aspect of the receiver concerns pumps and valves. The

receiver is the chamber in which the condensed vapor resides before it gets

pumped out. An electric pump is a transformer of electric power to mechanical

power to hydraulic power [36]. This is shown in the bond graph of Figure 2.10.

The exact nature of the electric power being supplied depends on the pump. A

pump is essentially a flow modulated pressure source. Various equations for a

pump illustrate this aspect [24, 51]:

∆P = kpump Qmax – q (2-72a)R

∆P = a + b q + c q2 (2-72b)GS

1

R:valve

SE SEPw1 Pw3

²P Qcw

Figure 2.9. Cooling water flow in condenser.

TM TMe Γorque ²Pi ω Q

Figure 2.10. Bond graph of pump.

59

The form of these equations is of a negative resistor. A valve is a hydraulic

resistor used to control the flow of the fluid. Equations for a valve include [24,

51, 52]:

∆P = kvalve 2 A2

q2 (2-73a)R

∆P = a + b q + c q2 + d q3 (2-73b)GS

Combining pumps and valves is the essence of hydraulic flow out of the

receiver and the base of the column. Fluid flow from the receiver first goes

through a pump, then is split into two pipes each containing a control valve.

One pipe, containing the reflux, leads to the stage plate below and the other

pipe, containing the distillate, leads out of the system. This setup is best shown

using a bond graph, Figure 2.11. The effort source on the right is the

01

1

1

R:pump R:valve1

R:valve2

SEP1 Pd1 PD

P2

qr + qd q

q

Ppmp Pv1

Pv2SE

PPE

distillate

reflux

Figure 2.11. Bond graph of pump and valves below receiver.

1

R:pump&valves

P1qd + qr

²P

P2

SEPD

SEPPE

qd + qr

I

01

1

1

R:pump R:valve1

R:valve2

SEP1 Pd1 PD

P2

qd

qr

Ppmp Pv1

Pv2

I

²P

²P

SE